---
tag: 04D7CHT5V1
kind: definition
name: Inverse hyperbolic functions
msc: [33B10]
defines: 0WFZX06698
uses: [0GWD21EWRW, 0EQFWKJ37J, 0MWYQTZ07Z, 0044CPHJBV, 0V7E9KP6Q2]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions
    locator: Definitions in terms of logarithms
---

## Definition

The inverse hyperbolic sine $\operatorname{arsinh} \colon \mathbb{R} \to \mathbb{R}$ is the inverse of [[$\sinh$|0GWD21EWRW]].
The inverse hyperbolic cosine $\operatorname{arcosh} \colon [1, \infty) \to [0, \infty)$ is the inverse of the restriction of [[$\cosh$|0EQFWKJ37J]] to $[0, \infty)$.
The inverse hyperbolic tangent $\operatorname{artanh} \colon (-1, 1) \to \mathbb{R}$ is the inverse of [[$\tanh$|0MWYQTZ07Z]].

## Remarks

Solving a quadratic equation in $e^y$ gives the formulas $\operatorname{arsinh} x = \ln(x + \sqrt{x^2 + 1})$ and $\operatorname{arcosh} x = \ln(x + \sqrt{x^2 - 1})$.
Similarly, $\operatorname{artanh} x = \frac{1}{2} \ln \frac{1 + x}{1 - x}$ for $|x| < 1$.
Their derivatives are $\frac{1}{\sqrt{x^2 + 1}}$, $\frac{1}{\sqrt{x^2 - 1}}$ for $x > 1$, and $\frac{1}{1 - x^2}$. Mathlib defines the first one as `Real.arsinh`.

## Explanations

### L1 human

The inverse hyperbolic functions undo the hyperbolic functions.
Since the hyperbolic functions come from $e^x$, their inverses come from the logarithm.